The violence also comes during a protracted political standoff that has raised sectarian tensions in a country racked by brutal communal bloodshed from 2006 to 2008.
Prime Minister Nuri al-Maliki and parliament speaker Osama al-Nujaifi both issued condemnations of the violence and appealed for calm, as did United Nations envoy Martin Kobler.
Monday's first attack struck at 11:00 am (0800 GMT) outside the Shiite endowment in Baab al-Muadham, in central Baghdad, and left at least 25 people dead and more than 65 wounded, medical officials said.
The bombing completely destroyed the endowment headquarters, its deputy chief, Sami al-Massudi, told AFP.
"We do not accuse anyone, but we call on the Iraqi people and especially on the sons of our religion to bury the strife because there is a plan to launch a civil war between the people, and between the Iraqi sects," Massudi said.
He said the Shiite endowment had received threats in recent days because of a dispute over the Al-Askari shrine, a Shiite mausoleum in the mostly Sunni city of Samarra.
The iconic gold-domed shrine was hit by a brutal Al-Qaeda suicide attack in February 2006 that ignited Iraq's bloody confessional violence.
Massudi and his aides had produced documents that attributed the management of the shrine to the Shiite religious endowment, sparking tensions with its Sunni counterpart.
"The issue of the Al-Askari shrine is a legal and constitutional issue, and it is our right, because it is a Shiite shrine," Massudi said.
The attack also fell on a significant day for Shiite Muslims -- the birthday of Imam Ali, a cousin and son-in-law of the Prophet Mohammed, who is a revered figure in Shiite Islam.
AFP journalists near the site of the attack said security forces cordoned off the area and barred anyone from approaching, while emergency workers searched for survivors in the remains of the endowment headquarters.
Several cars and nearby buildings were badly damaged by the explosion, and helicopters hovered overhead.
One man threw his spectacles to the ground and yelled: "They are all dead! They are all dead!"
Later on Monday, a statement on the Sunni endowment's website said that a mortar round struck near its headquarters in Adhamiyah, in north Baghdad. An interior ministry official said it was a roadside bomb that exploded near the building.
The attack did not cause any casualties, both said.
"We reject and condemn this criminal, cowardly, fanatical attack," Sunni endowment spokesman Faris al-Mehdawi said, referring to the attack against the Shiite endowment.
"These attacks aim to create divisions between the Iraqi people," he said. "There are dirty hands that are playing on sectarianism, and trying to bring the country back to the years of violence."
The latest bloodshed comes less than a week after a spate of bombings in Baghdad left 17 people dead on May 31, shattering a relative calm in the city.
The spike in attacks coincides with a ratcheting up of months-long tensions in which several political parties have called for the prime minister to be unseated.
"Maliki and Allawi are fighting over the government, and we are the victims," said Mohammed, who owns a restaurant opposite the Shiite foundation headquarters, referring to Iyad Allawi, a rival of the prime minister.
"Come and see the houses that were destroyed on the heads of children," he said, shirtless and covered in blood. "In my two hands, I carried children from under the rubble."
Monday's death toll was the highest from a single attack in Baghdad since a suicide bomber blew up a car outside a hospital on January 27 killing 31 people.
Violence in Iraq has declined dramatically since its peak in 2006-2007, but attacks remain common, especially in Baghdad. A total of 132 Iraqis were killed in violence in May, according to official figures.